{"id":428,"date":"2017-02-14T21:29:16","date_gmt":"2017-02-14T20:29:16","guid":{"rendered":"http:\/\/www2.mathnique.com\/site\/?page_id=428"},"modified":"2017-07-09T00:57:41","modified_gmt":"2017-07-08T22:57:41","slug":"cercles","status":"publish","type":"page","link":"https:\/\/www.mathnique.com\/site\/cercles\/","title":{"rendered":"Cercles"},"content":{"rendered":"<ul>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1629 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/travaux3-300x252.png\" alt=\"\" width=\"126\" height=\"106\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/travaux3-300x252.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/travaux3.png 718w\" sizes=\"auto, (max-width: 126px) 85vw, 126px\" \/><\/li>\n<li><span style=\"color: #ff0000;\"><strong>D\u00e9finition d'un cercle<\/strong><\/span><br \/>\nDans un plan affine euclidien , soit un r\u00e9el $R \\geq 0$ et un point $O$.<br \/>\nOn appelle cercle de centre $O$ et de rayon $R$ l'ensemble des points $M$ tels que $OM = R$<\/li>\n<li><strong><span style=\"color: #ff0000;\">Angles inscrits et angle au centre<\/span><\/strong><\/li>\n<li><span style=\"color: #ff0000;\"><strong>Corollaire : Cercle et angle droit<\/strong><\/span><br \/>\nSoit un cercle $\\mathcal{C}$ de diam\u00e8tre $[AB]$.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1982 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/cerclediametreAB-300x270.png\" alt=\"\" width=\"300\" height=\"270\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/cerclediametreAB-300x270.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/cerclediametreAB.png 364w\" sizes=\"auto, (max-width: 300px) 85vw, 300px\" \/><br \/>\n- Si $M \\in \\mathcal{C}$ alors le triangle $MAB$ est rectangle en $M$.<br \/>\non dit encore que tout triangle inscriptible dans un demi-cercle est rectangle.<br \/>\n- Si le triangle $MAB$ est rectangle en $M$ alors $M \\in \\mathcal{C}$.<\/li>\n<li><strong><span style=\"color: #ff0000;\">Autre Propri\u00e9t\u00e9 caract\u00e9ristique d'un cercle utilisant un diam\u00e8tre $[AB]$<\/span><\/strong><br \/>\n$M \\in \\mathcal{C}([AB]) \\iff\u00a0\\overrightarrow{MA}.\\overrightarrow{MB }= 0$<\/li>\n<li><strong><span style=\"color: #ff0000;\">Equation cart\u00e9sienne d'un cercle dans un rep\u00e8re orthonorm\u00e9<\/span><\/strong>\n<ul>\n<li><span style=\"color: #ff0000;\"><em>M\u00e9thode 1<\/em> <\/span>: $M(x;y) \\in \\mathcal{C}(\\Omega ; R) \\iff OM = R \\iff OM^2 = R^2 \\iff (x - x_{\\Omega})^2 +\u00a0(y - y_{\\Omega})^2 = R^2$<\/li>\n<li><em><span style=\"color: #ff0000;\">M\u00e9thode 2<\/span> <\/em>:\u00a0$M(x;y) \\in \\mathcal{C}([AB]) \\iff\u00a0\\overrightarrow{MA}.\\overrightarrow{MB }= 0$<br \/>\n$\\iff (x_A - x)(x_B - x) +\u00a0(y_A - y)(y_B - y) = 0$<br \/>\n$\\iff x^2 + y^2 -x(x_A + x_B) - y(y_A + y_B) +x_A x_B +y_A y_B = 0$<\/li>\n<\/ul>\n<\/li>\n<li><strong><span style=\"color: #ff0000;\">Probl\u00e8me r\u00e9ciproque : Qu'est ce que l'ensemble $\\Gamma$ des points $M$ v\u00e9rifiant une \u00e9quation cart\u00e9sienne du type<\/span>\u00a0<span style=\"color: #ff0000;\">$x^2 + y^2 + ax + by + c = 0$ dans un rep\u00e8re orthonorm\u00e9 ?<br \/>\n<\/span><\/strong>$M(x;y) \\in \\Gamma \\iff\u00a0x^2 + y^2 + ax + by + c = 0$<br \/>\n$\\iff (x \\ - \\dfrac{a}{2})^2 + (y \\ - \\dfrac{b}{2})^2 = c \\ - \u00a0\\dfrac{a^2}{4} \\ - \\dfrac{b^2}{4} $- ou bien $ c \\ - \u00a0\\dfrac{a^2}{4} \\ - \\dfrac{b^2}{4}&lt; 0$ alors $\\Gamma =\\emptyset$- ou bien $ c \\ - \u00a0\\dfrac{a^2}{4} \\ - \\dfrac{b^2}{4}= 0$ alors $\\Gamma =\\{\\Omega(\\dfrac{a}{2} ; \\dfrac{b}{2})\\}$ = le cercle-point $\\Omega$<\/p>\n<p>- ou bien $ c \\ - \u00a0\\dfrac{a^2}{4} \\ - \\dfrac{b^2}{4}&gt; 0$<br \/>\nalors $\\Gamma =$ le cercle $\\mathcal{C}$ de centre $\\Omega(\\dfrac{a}{2} ; \\dfrac{b}{2})$ et de rayon $ \\sqrt{c \\ - \u00a0\\dfrac{a^2}{4} \\ - \\dfrac{b^2}{4}}$<\/li>\n<li>Syst\u00e8mes d'\u00e9quations param\u00e9triques d'un cercle<\/li>\n<li><\/li>\n<li><strong><span style=\"color: #ff0000;\">Cercle capable ou Cercle d'APPOLONIUS associ\u00e9 \u00e0 deux points $A,B$ et un angle $\\theta$<\/span><\/strong><\/li>\n<li><strong><span style=\"color: #ff0000;\">Cercle d'Euler<\/span><\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1463 aligncenter\" src=\"http:\/\/www2.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/euler.png\" alt=\"\" width=\"113\" height=\"122\" \/><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>D\u00e9finition d'un cercle Dans un plan affine euclidien , soit un r\u00e9el $R \\geq 0$ et un point $O$. On appelle cercle de centre $O$ et de rayon $R$ l'ensemble des points $M$ tels que $OM = R$ Angles inscrits et angle au centre Corollaire : Cercle et angle droit Soit un cercle $\\mathcal{C}$ de &hellip; <a href=\"https:\/\/www.mathnique.com\/site\/cercles\/\" class=\"more-link\">Continuer la lecture<span class=\"screen-reader-text\"> de &laquo;&nbsp;Cercles&nbsp;&raquo;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-428","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages\/428","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/comments?post=428"}],"version-history":[{"count":26,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages\/428\/revisions"}],"predecessor-version":[{"id":2000,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages\/428\/revisions\/2000"}],"wp:attachment":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/media?parent=428"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}